HYACINTHOS 25266

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Labc = the reflection of AB in AC

Ca = the orthogonal projection of C on Labc

Lacb = the reflection of AC in AB
Ba = the orthogonal projection of B on Lacb

Similarly:

Lbca = the reflection of BC in BA
Ab = the orthogonal projection of A on Lbca

Lbac = the reflection of BA in BC
Cb = the orthogonal projection of C on Lbac

and

Lcab = the reflection of CA in CB
Bc = the orthogonal projection of B on Lcab

Lcba = the reflection of CB in CA
Ac = the orthogonal projection of A on Lcba

Ma, Mb, Mc = the midpoints of BaCa, CbAb, AcBc, resp.

The NPCs of ABC, MaBC, MbCA, McAB are concurrent.

[Angel Montesdeoca]:

The NPCs of ABC, MaBC, MbCA, McAB are concurrent  at R

R = ( (b^2-c^2)^2 (2 a^8
                         -7 a^6 (b^2+c^2)
                         +a^4 (9 b^4+4 b^2 c^2+9 c^4)
                         +a^2 (-5 b^6+8 b^4 c^2+8 b^2 c^4-5 c^6)
                         +(b^2-c^2)^2 (b^4-3 b^2 c^2+c^4)) : ... : ... ),
                        
             with search number     2.887616366735193662803171
             R lies on lines :   {113,3850},  {114,10276},  {128,10277},  {137,7668},  ....
            
  Angel Montesdeoca